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%def\bps (vedi sotto)
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%%%%%%%%%
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%%%%%%%%%%%% Definizioni numeri teoremi %%%%%%%%%%
\def\dmevvf{4.1}
\def\dmopva{4.2}
\def\dmufre{3.1}
\def\dspecf{3.4}
\def\eopza{2.1}
\def\efffat{5.7}
\def\lawmev{5.1}
\def\lfffat{5.7}
\def\lfopmt{5.6}
\def\lgrcom{5.5}
\def\lirrap{5.3}
\def\lmeapr{5.4}
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\def\pcnsms{4.3}
\def\rrespe{2.2}
\def\tchrep{4.4}
\def\peqsps{3.2}
\def\tspmul{3.3}
%%%%%%%%% references %%%%%%%%%%%
\def\amst{1}
\def\bmsp{2}
\def\brro{3}
\def\bufr{4}
\def\cath{5}
\def\cava{6}
\def\dunf{7}
\def\fave{8}
\def\hiph{9}
\def\kehl{10}
\def\klsh{11}
\def\loms{12}
\def\manu{13}
\def\maur{14}
\def\mcsh{15}
\def\nath{16}
\def\nicl{17}
\def\seie{18}
\def\vnbo{19}
\def\vnth{20}
\def\vnus{21}
\def\zzak{22}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$\ $
\rc
{\rmm A Generalization of the Stone-von Neumann Theorem}
\pn
{\rmm to Non-Regular Representations of the CCR-Algebra}
\rc\rc
S. CAVALLARO${}^1$, G. MORCHIO${}^2$ and F. STROCCHI${}^3$
\pn
{\ipic ${}^1$Dipartimento di Matematica e Informatica
dell'Universit\`a, Udine, Italy
\pn
${}^2$Dipartimento di Fisica dell'Universit\`a and INFN, Pisa, Italy
\pn
${}^3$Scuola Normale Superiore and INFN, Pisa, Italy}
\rc\rc
{\bpic Abstract.} {\pic We give a classification, up to unitary
equivalence, of the
representations of the $C^*$-algebra of the Canonical
Commutation Relations which generalizes the classical
Stone-von Neumann Theorem to the case of representations
which are strongly measurable, but not necessarily
strongly continuous. The classification includes all
the (non-regular) representations which have been considered in
physical models.}
\rc\rc
{\bf 1. Introduction}
\rc
The canonical approach to the description
of quantum systems is based on the
use of the so-called canonical variables $\{\hat q, \hat p\}$ satisfying
the Heisenberg commutation relations,
in terms of which all the physical quantities, including
the Hamiltonian, are defined. From a mathematical point of view it is more
convenient to deal with the so-called {\it Weyl operators}, i.e.
the unitary operators
$e^{-i(a\hat q+b\hat p)}$, because they are bounded and
can be considered as representatives of elements of an abstract
$C^*$-algebra, the {\it algebra of the Canonical Commutation
Relations} or, briefly, the {\it CCR-algebra}.
Furthermore the Weyl operators have the advantage of being meaningful
also in the cases in which the corresponding generators $\hat q, \hat p$
cannot be recovered as self-adjoint operators in the Hilbert space
of the states of the quantum system (non-regular representations
of the CCR-algebra) and the use of $\hat q$ and
$\hat p$ can only be justified
at an heuristic level, e.g.
in order to define the Hamiltonian.
\par
These cases, which were not considered in the foundations of
quantum mechanics [\vnbo], probably because they were regarded
as unphysical pathologies, have recently turned out to be of physical
interest to treat quantum systems whose ground state
formally corresponds to a non-$\lq$ function. This phenomenon
occurs when there are superselection sectors,
in particular when the system lives
on a compact manifold (quantum particle on a circle [\amst]),
when the invariance under a ``gauge'' group characterizes the ground
states (quantum particle in a periodic potential, the Bloch
electrons [\amst,\loms]), when a magnetic field gives rise to
non-commutative translations and their eigenstates
are considered [\zzak].
This phenomenon also occurs in two-dimensional QFT
models (massless fields in $1+1$ dimensions [\amst]), in the
temporal gauge formulations of gauge QFT
[\bufr, \nath], in the positive gauge quantization of
Chern-Simons theory [\nicl] and it is at the basis of the so-called
$\theta$-vacua superselection sectors [\amst, \loms].
\par
The above considerations suggest to reconsider the classical
Stone-von Neumann (SvN) Theorem [\vnth], which gives the uniqueness
(up to unitary equivalence)
of the regular irreducible representation of the CCR-algebra,
and to look for a classification
of non-strongly continuous representations, covering
the non-regular ones needed for physically interesting models.
%
\rc\rc
%
{\bf 2. Regular Representations and Zak Transform}
\rc
For a quantum system with one degree of freedom
the {\it algebra of the Canonical Commutation
Relations} is, by definition, the
$C^*$-algebra generated
by the abstract elements, $W(a,b)$, $(a,b)\in \R^2$, with product rule
and involution given by
$$W(a,b)\ W(c,d)\,=\, e^{-{i\over 2}(ad-bc)}\ W(a+c,b+d)
\qquad\qquad
\bigl (W(a,b)\bigr )^*\,=\, W(-a,-b)$$
(see [\brro, \manu] for more details).
We shall denote this algebra by $\ccr$.
A fundamental result in the study and classification
of the representations of $\ccr$ is the
\rb
STONE-VON NEUMANN UNIQUENESS THEOREM.
{\it Let $(\h ,\pi)$ be a
representation of $\ccr$ satisfying the hypotheses:}\pn
\item{(a)}{\it $(\h ,\pi)$ is nondegenerate and
irreducible}
\item{(b)}{\it the operator-valued function
$(a,b)\to\pi(W(a,b))$
is strongly continuous.}\pn
{\it Then $(\h ,\pi)$ is unitary equivalent to the
Schr\"odinger representation.}
\rb
A representation of $\ccr$ which satisfies the property (b)
of the theorem is called {\it regular}.
\par
The Schr\"odinger representation $\pi_{S}$
is based on the realization of
the maximal abelian subalgebra generated by $W(0,b)$, $b\in \R$,
as multiplication operators in $\lq(\R)$, where $\R$ arises
as the spectrum of the (unbounded) generator of the
strongly continuous group $\pi_{S}(W(0,b))$.
More convenient unitarily equivalent realizations
({\it Zak representations})
are obtained by choosing as maximal abelian
$C^*$-subalgebras those generated by two elements,
$W(\lambda,0)$ and $W(0,2\pi/\lambda)$, for a fixed
positive $\lambda$. The advantage is that such
sub-algebras are finitely generated, their Gelfand
spectrum is the two-dimensional torus and, under the
assumptions of the SvN Theorem, they
can be realized as multiplicative operators in the
space of square-integrable functions on their spectrum
with the Lebesgue measure. Such subalgebras
also play a relevant r\^ole in the above models.
Up to automorphisms one can chose $\lambda=1$; we shall denote
by $\zac$ the resulting $C^*$-subalgebra and identify the
Gelfand spectrum of $\zac$ with the product space
$\T=[0,1)\times [0,2\pi)$.
The isometric isomorphism $\U_Z$ ({\it Zak transform}
[\zzak,\klsh]) from the Schr\"odinger representation
to the $\lambda=1$ Zak representation is given by
$$\eqalign{
&\bigl (\U_Z\,\phi\bigr )(\a,\b)
=\sum_{n\in\Z}\phi(n+\a)\ e^{-i\, n\b}
\qquad\qquad\qquad\qquad\ \forall\phi\in \lq(\R)\cr
&\bigl({\U}_Z^{-1}\,\psi\bigr )(x)={1\over{2\pi}}\int_0^{2\pi}\!\psi
(x\, \mod 1, \b)\ e^{i\, [x]\b}\, \d \b\qquad \qquad
\forall\psi\in \lq (\T\!,{1\over{2\pi}}\d \a\, \d\b)\cr}$$
where $x\in \R$, $(\a,\b)\in [0,1)\times [0,2\pi)$ and
$[x]$ is the integer part of $x$.
>From these relations it follows that, for each $a,b$ in $\R$ and
$\psi$ in $\lq (\T\!,{1\over{2\pi}}\d \a\, \d\b)$,
%\def\eopza{2.1}
$$\eqalign{
&\bigl(\U_Z\; \pi_{S} (W(a,0))\;\U_Z^{-1}\,\psi\bigr)(\a,\b)\,=\,
e^{i\,[\a+a]\b}\; \psi((\a+a)\,\mod 1,\b)\cr
&\bigl(\U_Z\; \pi_{S} (W(0,b))\;\U_Z^{-1}\,\psi\bigr)(\a,\b)\,=\,
e^{-i\,b\a}\; \psi(\a,(\b+b)\mod\, 2\pi)\quad .\cr}\eqno{(\eopza)}$$
In conclusion, a regular irreducible and
nondegenerate representation of $\ccr$
is unitarily equivalent to a representation on the space
$\lq (\T\!,{1\over{2\pi}}\d \a\, \d\b)$ such that the elements of
the abelian $C^*$-subalgebra $\zac$ act
multiplicatively by their Gelfand transforms.
%
\rc\rc
%
{\bf 3. Spectrally Multiplicity-Free Representations}
\rc
In the following our strategy will be to characterize the representations
of $\ccr$ in terms of representations of its maximal abelian
$C^*$-subalgebra $\zac$. The first problem is to understand under which
condition the representation of $\zac$
can be described in terms of a space
$\lq(\T\!, \mu)$, where $\mu$ is a positive measure on the Borel
(equivalently Baire) sets of $\T$,
and the elements of $\zac$ act as multiplicative operators by
their Gelfand transforms.
This condition is general enough to cover all the above mentioned models
and, in our opinion, it represents a mild restriction which allows
a measure theoretical discussion of the problem.
In the separable case such a property amounts to the standard
property of multiplicity-free, which in turn follows from
the irreducibility and regularity of the representation of $\ccr$.
However, in the non-separable case, which is typical of
non-regular representations of $\ccr$, a multiplicity-free
representation of an abelian unital $C^*$-algebra $\ba$
cannot, in general, be diagonalized on a single copy of
its spectrum (see [\cath, \cava] for a counterexample).
More precisely, in the following, we denote by $\ba$
an abelian unital $C^*$-algebra and by $(\h, \pi)$ a representation of $\ba$;
for every vector $x$ in $\h$, we
denote by $\h_x$ the cyclic $\pi(\ba)$-invariant
subspace of $\h$ given by the closure of the set
$\{\pi(A)x\,|\,A\in \ba\}$ and by
$\mu_x$ the positive Baire measure on the
Gelfand spectrum $\spb$ of $\ba$
defined by $x$
via the Riesz--Markov Representation Theorem
$$(x\, , \,\pi (A)x)\, =\int _{\spb}\ta\, \d\mu_{x}
\qquad \quad \forall\, A\in \ba $$
(with $\ta$ the Gelfand transform of $A$).
\rb
%\def\dmufre{{3.1}}
DEFINITION \dmufre. A nondegenerate representation $(\h,\pi)$
of $\ba$ is {\it multiplicity-free}
if it verifies one
(and therefore all) of the following equivalent [\cath,\cava]
conditions
\item{\rm (i)} for each vector $x$ in $\h$, the projection on the
cyclic subspace $\h_x$ belongs to the von Neumann algebra,
$\pi(\ba)''$,
generated by $\pi(\ba)$
\item{\rm (ii)} $\pi(\ba)''$ is maximal abelian
\item{\rm (iii)} if $x,y\in\h$ and $\h_x\perp\h_y$, then
$\mu_x$ and $\mu_y$ are mutually orthogonal, i.e., for every $x$ in $\h$
and every vector $y\perp \h_x$,
there exists a Baire set, $S_x^y$ of $\spb$ such that
$\mu_x(\spb\ \backslash\, S_x^y)=0$ and $\mu_y(S_x^y)=0$.
\rb
In separable Hilbert spaces we have [\maur]:
\rb
%\def\peqsps{{3.2}}
PROPOSITION \peqsps.
{\it Let $(\h,\pi)$ be a multiplicity-free
nondegenerate representation of $\ba$, with $\h$ a
separable Hilbert space. Then
there exists a positive measure $\mu$ on the Baire
sets of $\spb$ and
an isometric isomorphism, $U$, from $\h$ onto $\lq (\spb ,\mu)$
such that, for each
element $A$ of $\ba$, $U\pi (A)\,U^{-1}$ is the
operator of multiplication by the Gelfand transform
of $A$.}
\rb
In the non-separable case, the r\^ole of the von Neumann algebra
$\pi(\ba)''$
is played [\cath, \cava] by
the Baire*-algebra generated by
$\pi(\ba)$ (i.e. the smallest $C^*$-algebra of operators
in $\h$ which contains $\pi(\ba)$ and the limit
of each of its weakly convergent monotone sequences [\kehl]):
\rb
%\def\tspmul{3.3}
THEOREM \tspmul. {\it Let $(\h,\pi)$ be a
representation of a unital abelian $C^*$-algebra $\ba$.
Then the following statements are equivalent:}
\pn
\item{\rm (i)} {\it for each vector $x$ in $\h$, the projection on the
cyclic subspace $\h_x$ belongs to the Baire*-algebra
generated by $\pi(\ba)$}
\item{\rm (ii)} {\it for every $x$ in $\h$, there exists a Baire subset,
$S_x$, of $\spb$ such that
$\mu_x(\spb\ \backslash\, S_x)=0$ and
$\mu_y(S_x)=0$ for each vector $y\perp \h_x$
\item{\rm (iii)} there exist a positive measure $\mu$ on the Baire
sets of $\spb$ and
an isometric isomorphism $U$ from $\h$ onto $\lq (\spb ,\mu)$
such that, for each
element $A$ of $\ba$, $U\pi (A)\,U^{-1}$ is the
operator of multiplication by the Gelfand transform of $A$.}
\rb
%\def\dspecf{3.4}
DEFINITION \dspecf. We say that a representation of
$\ba$ is {\it spectrally multiplicity-free} if it verifies
one (and therefore all) of the statements of Theorem \tspmul.
\rb
Finally, if $(\h,\pi)$ is spectrally multiplicity-free, the measure $\mu$,
in the statement (iii) of Theorem \tspmul,
can be explicitly constructed as the sum
$$\mu\,=\, \sum_{i\in I}\,\mu_{x_i}$$
where ${\{x_i\}}_{i\in I}$ is a family of orthogonal vectors
such that $\h=\oplus_{i\in I}\h_{x_i}$.
\par
According to the above discussion, in the following we shall
consider those representations of $\ccr$ whose restriction to
the commutative subalgebra $\zac$ is spectrally
multiplicity-free.
This property is related to the irreducibility of the
representation of the full algebra $\ccr$; it is in fact equivalent
to it in the regular case and it implies irreducibility
under measurability conditions discussed below.
%
\rc\rc
%
{\bf 4. A Classification of Strongly Measurable
Representations of $\ccr$}
\rc
Also within the class of spectrally
multiplicity-free representations of $\zac$
pathologies may arise
if no ``regularity'' condition is required on the representation
of the full algebra $\ccr$. For instance,
even in the separable case, one may construct (multiplicity
free) representations $\pi$
which are not weakly measurable in the parameters
of the Weyl operators
$\pi(W(a,b))$ [\cath; Ex. V.3.2], so that even the Borel
structure of $\R^2$ may be lost
in the representation, excluding a discussion in terms
of the Borel measures of Theorem \tspmul. We recall that,
for representations in separable Hilbert
spaces, the strong continuity of the Weyl operators (as function from $\R^2$
into $\bh$) is equivalent to their strong, or weak, measurability [\vnus].
The SvN Theorem thus amounts to a classification of strongly measurable
representations in separable spaces. To obtain a generalization
to non-regular representations we are led to discuss
measurability properties in non-separable spaces.
\par
Let $(X, \M, \mu)$ be a positive $\sigma$-finite measure space and
$\h$ be an Hilbert space.
\rb
%\def\dmevvf{4.1}
DEFINITION \dmevvf. A function
from $X$ into $\h$ is said to be {\it countably-valued} if it assumes at
most a countable set of values in $\h$, each value being taken on a
measurable set.
A function from $X$ into $\h$ is called
{\it measurable w.r.t. $\!\mu$} if there
exists a sequence of countably-valued functions converging
$\mu$-almost everywhere to it.
\rb
%\def\dmopva{4.2}
DEFINITION \dmopva. An operator-valued function $V$ from $X$
into the set $\bh$ of bounded operators in $\h$
is called {\it strongly measurable w.r.t. $\!\mu$} if,
for every $x$ in $\h$, the vector-valued function
$X\ni a\to V(a)x\in \h$ is measurable w.r.t. $\mu$.
\rb
%\def\pcnsms{4.3}
PROPOSITION \pcnsms. [\hiph; Thm. 3.5.5]
{\it An operator-valued function $V:X\to \bh$
is strongly measurable w.r.t. $\!\mu$ iff:}
\item {(i)} {\it it is weakly measurable (i.e. $\forall x, y\in \h$,
the function $X\ni a\to(V(a)x\,|\, y)\in \C$ is measurable)}
\item {(ii)} {\it $V(a)x$ is $\mu$-almost
separably-valued for every $x$ in $\h$ ( i.e. for every $x$ in $\h$,
there exists a $\mu$-null set $N$ such that
$\{V(a)x\,|\, a\in X\backslash N\}$ is a separable subset of $\h$).}
\rb\par
By the discussion at the beginning of the section, we have to require
some measurability property of the operator-valued function
$\R^2\ni (a,b)\rightarrow\pi(W(a,b))$.
Actually, since in our analysis the Zak algebra is already represented
by multiplication operators, it is enough to discuss the
measurability of $\pi(W(a,b))$ with $(a,b)
\in [0,1)\!\times\![0,2\pi)$.
Since the adjoint action of $W(a,b)$ on $\zac$ induces
a translation on the spectrum of $\zac$, it is natural to identify
the parameter space $[0,1)\!\times\![0,2\pi)$ with the spectrum
$\T$ of $\zac$ and to require strong measurability w.r.t. every measure
$\mu_x$, $x\in \h$.
\par
For each vector $y$ in $\h$, the set
$\{\pi(W(a,b))y\,|\, (a,b)\in\T\}$ is, in general, non-separable, but
strong measurability guarantees a sort of ``local separability''
of the representation; in fact, by Proposition \pcnsms,
it implies that, for every measure $\mu_x$,
there exists a Borel subset $N$ of $\T$ such that $\mu_x(N)=0$ and
$\{\pi(W(a,b))y\,|\, (a,b)\in\T\,\backslash\, N\}$ is separable.
This will allow to make a ``local'' use of standard results
which hold only in separable Hilbert spaces
(or for $\sigma$-finite measures),
even if the representation is non-separable.
We can now state our theorem.
\rb
%\def\tchrep{4.4}
THEOREM \tchrep. {\it Let $(\h, \pi)$ be a representation of the
CCR-algebra $\ccr$ satisfying the following hypotheses:}
\item{(A)} {\it spectrally multiplicity-free condition
for the subalgebra $\zac$}
\item{(B)} {\it strong measurability w.r.t. every measure $\mu_x$,
$x\in \h$, i.e.}
$$[0,1)\!\times\![0,2\pi)\ni (a,b)\, \longrightarrow\,
\pi(W(a,b))y\in \h$$
\item{}{\it is $\mu_x$-measurable for all $x$ and $y$ in $\h$.}
\pn
{\it Then:}
\item{(1)} {\it $(\h, \pi)$ is an irreducible representation of $\ccr$}
\item{(2)} {\it there exist a positive translation-invariant
measure $\mu$ on the Borel $\sigma$-algebra of $\T$ and
an isometric isomorphism $\U$ from $\h$ onto $\lq(\T\!,\mu)$ such
that, for every $a,b$ in $\R$ and every $\psi$ in $\lq(\T\!, \mu)$,
$$\eqalign{
&\bigl (\U\; \pi (W(a,0))\;
\U^{-1}\;\psi\,\bigr )(\a,\b)\,=\,
e^{i\,[\a+a]\b}\ \, \psi((\a+a)\,\mod 1,\b)\cr
&\bigl( \U\; \pi (W(0,b))\;
\U^{-1}\;\psi\,\bigr )(\a,\b)\,=\,
e^{-i\,b\a}\ \, \psi(\a,(\b+b)\,\mod 2\pi)\cr}\eqno{(4.1)}$$
where $[\a+a]$ denotes the integer part of $\a+a$}
\item{(3)} {\it the measure $\mu$ can be written
as the sum of a family of finite positive Borel
measures concentrated on disjoint sets.}
\rb
Conditions (A) and (B) allow to characterize the representations,
up to unitary equivalence: inequivalent
representations correspond to inequivalent
translation-invariant measures.
This classification covers the known examples:
\pn
\item {$-$} the Schr\"odinger representation corresponds
to the two-dimensional Lebesgue measure on $\T$; hence
$\h\cong \lq (\T\!,{1\over{2\pi}}\d \a\, \d\b)$
\pn
\item {$-$} the (equivalent) representations defined, via the
GNS construction, by the ``momentum states'' $\omega_p\,$, $p\in \R$,
$$\omega_p(W(a,b))=\cases{
0\;, & if $b\ne 0$ \cr
e^{i\, pa}\;, & if $b=0$ \cr}$$
(see [\amst, \bmsp, \fave]), correspond to the measure
${\mu} = \sum_{j\in [0,2\pi)}\d\a_j$ where $\d\a_j\,$, $j\in [0,2\pi)$,
denotes the one-dimensional Lebesgue measure
concentrated on the interval $\{(\a,j)\,|\, \a\in [0,1)\}\subseteq\T$
\pn
\item {$-$} the (equivalent) representations defined by the ``Zak states''
$\omega_{\zeta \gamma}\,$, $\zeta\in [0, 2\pi)$, $\gamma\in [0,1)$,
$$\omega_{\zeta \gamma}(W(a,b))=\cases{
0\;, & if $(a,b)\not\in\Z\times 2\pi\Z $ \cr
e^{i\pi m\, n}\; e^{i\, n\zeta}\; e^{i2\pi m\,\gamma}\;,
& if $(a,b)=(n,2\pi m)$ \cr}$$
(see [\bmsp,\zzak]), correspond to the counting measure
on $\T$; hence $\h\cong {\ell}^2(\T)$.
%
\rc\rc
%
{\bf 5. Proof of Theorem \tchrep}
\rc
The logic of the proof is the following. By exploiting the
spectrally multiplicity-free condition and the strong measurability
we first show that the representation is irreducible (Lemma \lirrap).
Then we reduce the analysis of the representation to that of a unitary
representation $T$ of the commutative group $\tor$
(Lemma \lgrcom).
Such representation is shown to be of the form
$$\bigl (T(\o a)\psi\bigr )(\o \a)\, =
\, f(\o a,\o \a)\; \psi(\o \a+\o a)$$
(Lemma \lfopmt). By exploiting the measurability of
the function $f$, which follows from the hypothesis (B), and the
group law, $f$ is shown to be of the form
$$f(\o a,\o \a)\;=\;{{\xi(\o a+\o \a)}\over{\xi(\o \a)}}$$
(Lemma \lfffat). Hence $f$ can be removed by a unitary transformation
and the relations (4.1) are easily obtained.
\par
Let $(\h,\pi)$ be a representation of $\ccr$ satisfying the
hypotheses (A) and (B) of the theorem.
According to the spectrally multiplicity-free property
(see Section 3), there exist
a positive measure $\mu$ on the Borel $\sigma$-algebra $\bs$
of $\T$ and an isometric isomorphism $\U$ from
the representation space $\h$ onto
$\lq(\T, \mu)$ such that, for each $A$ in $\zac$,
$\U\,\pi (A)\,\U^{-1}$ is the operator of multiplication by
the Gelfand transform of $A$.
The measure $\mu$ is of the form
$\mu=\sum_{i\in I}\mu_{x_i}$
where $\{x_i\}_{i\in I}$ is a family of orthogonal vectors
such that $\h=\oplus_{i\in I} \overline {\{\pi(\zac)x_i\}}$
and $\mu_{x_i}$ is the measure associated to $x_i$ via
the Riesz-Markov Theorem.
Moreover, for each $i$ in $I$, there is a Borel
set $S_i$ of $\T$ such that $\mxi(S_i)=\mxi(\T)$ and
$\mu_y(S_i)=0$ for all $y\perp \h_{x_i}$.
\pn
If $f$ is a function on $\T$ and $(a,b) \in \R^2$, denote
by $f^{(a,b)}$ the translate of $f$, i.e.
$f^{(a,b)}(\a,\b)=f ( (\a-a)\,\mod 1,(\b-b)\,\mod 2\pi)$.
If $\mu$ is a positive measure on $\bs$, its
translate, $\mu^{(a,b)}$, is the measure
defined by $\mu^{(a,b)}(f)=\mu(f^{(-a,-b)})$,
for every bounded Borel-measurable function $f$ on $\T$.
\rb
%\def\lawmev{{5.1}}
LEMMA \lawmev. {\it For every $(a,b)$ in $\R^2$ and $x$ in $\h$, the
measure associated to the
vector $\pi(W(a,b))x$ is the translated measure $\mu_x^{(-a,-b)}$.}
\pf
The relation $\tau_{ab}(A)=W(a,b)^{-1}A \;W(a,b)$, $ A\in\zac$, defines
a inner automorphism of $\zac$.
This map induces, by the Gelfand transform,
an automorphism $\widehat{\tau_{ab}}$ on $\con(\T)$:
$\widehat{\tau_{ab}}(\ta\,)\,=\,\ta^{(a,b)} $.
On the other hand, from the definition of $\mu_x$ one has, for every
$\ta$ in $\con(\T)$,
$$\int_{\T}\widehat{\tau_{ab}}(\ta\,)\ \d\mu_x\ =
\bigl ( x \, , \, \pi\bigl (W(a,b)^{-1}A\ W(a,b) \bigr )
x \bigr ) \ = \ \int_{\T}\ta \ \d \mu_{{\pi (W(a,b) )x}} \ \ .
$$
\qed
In the following, for simplicity, a
point of $\T$ will be denoted by an overlined
(Greek or Latin) letter; $\o \a + \o a \,$ will indicate a
translation of $\o \a \in \T$ by $\o a \in \T$.
For each pair $(i,j)$ of indices in $I$ one can consider
the product measure $\mxi\ot\mxj$ defined
on the Borel $\sigma$-algebra $\bps$ of the product space $\T\times\T$.
\rb
%\def\lpmipr{5.2}
LEMMA \lpmipr. {\it Let $(i,j)$ be a pair of indices in $I$
and let $A$ be an element of the $\sigma$-algebra $\bps$
such that $(\mxi\ot\mxj)(A)>0$. Then there
exists $k$ in $I$ such that}
$$\bigl (\mxi\ot\mxj\bigr )(A\cap \Im_k)>0\ ,\qquad
\hbox{{\it with}} \quad \Im_k= \bigl \{(\o a,\o \a)\in
\T\!\times\!\T\,\bigl |\, \chi_{S_k}(\o \a+\o a)=1\bigr\}\ \ .$$
\pf
Note firstly that $\Im_k\in \bps$ (in fact, if $\eta$ is the
homeomorphism of $\T\times\T$ such that
$(\o a\, ,\o \a)\tends {\eta}(\o a\, ,\o \a-\o a)$, then
$\Im_k=\eta(\T\times S_k)$).
Let $A$ be an element of $\bps$ such that $(\mxi\ot\mxj)(A)>0$.
We can always assume that $A\subseteq S_i\times S_j$.
By Fubini Theorem there exists a measurable
$Y_A \subseteq S_i$, with $\mxi (Y_A)>0$, so that,
for every $\o a$ in $Y_A$, the set
$$A|_{\o a} \,=\, \{\o \a\in \T\,|\,(\o a,\o \a)\in A\}$$
(the $\o a$-section of $A$) is Borel measurable and $\mxj(A|_{\o a})>0$.
Thus, if $k$ is an index in $I$ for which
$(\mxi\ot\mxj) (A\cap \Im_k)=0$,
since $A|_{\o a}\subseteq S_j\,$,
$$\int_{\T}\chi_{A|_{\o a}}(\o \a)\
\chi_{S_k}(\o \a+\o a)\ \d\mu(\o \a)\,=\,0
\qquad\qquad \mxi(\o a)\hbox{-a.e.}\ .$$
Using the translation property of the measures $\mu_x$ (Lemma \lawmev),
for every $\tb $ in $\con (\T)$,
$$\eqalign{
\bigl (\,&\chi_{A|_{\o a}}\; , \;\U\pi(W(\o a))\U^{-1}
\tb\chi_{S_k}\,\bigr )\cr
&\ =\!\int_{\T}\!\bigl (\U\pi(W(\o a))\U^{-1}\,\tb\chi_{S_k}\bigr )(\o \a)\ \,
\chi_{S_k}\!(\o \a+\o a)\ \,
\chi_{A|_{\o a}}\!(\o \a)\;\d\mu(\o \a)\,=\,0
\qquad\mxi(\o a)\hbox{-a.e.}\ .\cr}$$
Therefore $(\mxi\ot\mxj) (A\cap \Im_k)=0$ implies
$\pi(W(-\o a))\,\U^{-1}\chi_{A|_{\o a}}\,\perp\,\h_{x_k}$
almost everywhere w.r.t. $\mxi(\o a)$.
By the measurability property (B), one has that,
removing at most a $\mxi$-null set from $Y_A$, the set
$\{\pi(W(-\o a))\, \U^{-1} \chi_{A|_{\o a}}\,|\, \o a\in Y_A\}$
is separable. Since the subspaces $\h_{x_k}$ are mutually orthogonal,
this implies that the family
$$\widetilde I\,=\,\Bigl \{k\in I\;\Bigl |\; \h_{x_k}\; \not\perp\;
\pi(W(-\o a))\,\U^{-1}\chi_{A|_{\o a}}\ \ \
\hbox{for some} \ \o a\in Y_A\Bigr \}$$
must be countable.
In conclusion, if $(\mxi\ot\mxj) (A\cap \Im_k)=0$,
for every $k$ in $I$, then,
since $\widetilde I$ is countable,
$$\pi(W(-\o a))\,\U^{-1}\chi_{A|_{\o a}}\ \,
\perp\ \,\oplus_{k\in\widetilde I}\;
\h_{x_k}\qquad\qquad\mxi(\o a)\hbox{-a.e.}\ .$$
So $\pi(W(-\o a))\,\U^{-1}\chi_{A|_{\o a}}$ is zero
$\mxi (\o a)$-a.e..
Since $\pi(W(-\o a))\,\U^{-1}$ is norm-preserving, this means
that $\mxj(A|_{\o a})=0$ for $\mu_{x_i}$-almost all $\o a$ in $Y_A$,
which contradicts the property previously obtained for $Y_A$.\qed
\rb
%\def\lirrap{5.3}
LEMMA \lirrap. {\it The representation $(\h ,\pi)$ is irreducible.}
\pf
Let $x_1$ be a non-null vector of $\h$ and
$\h_1=\overline {\{\pi(\ccr)x_1\}}$. Let $x_2\perp \h_1$.
Consider $\h_2=\overline {\{\pi(\ccr)x_2\}}$ and the
subrepresentation $(\h_1\oplus\h_2, \pi_1\oplus\pi_2)$.
Writing
$$\h_1=\oplus_{l\in I_1}\, \overline {\{\pi(\zac)x_{1,l}\}}
\quad , \qquad\qquad
\h_2=\oplus_{m\in I_2}\, \overline {\{\pi(\zac)x_{2,m}\}}$$
we obtain the decomposition $\h_1\oplus \h_2
=\oplus_{j=1,2\ m\in I_j}\, \h_{x_{j,m}}$.
Now for every product measure $\mu_{x_{1,l}}\ot\mu_{x_{2,m}}$,
with $l$ in $I_1$ and $m$ in $I_2$, and every $k$ in $I_1$,
$$(\mu_{x_{1,l}}\ot\mu_{x_{2,m}})(\Im_{1,k})\,=\,
\int_{\T}\mu_{x_{2,m}}({S_{1,k}}^{(-\o a)})\ \,\d\mu_{x_{1,l}}(\o a)\,=\,0$$
(${S_{1,k}}^{(-\o a)}$ is the $(-\o a)$-translated of $S_{1,k}$),
since $x_{1,k}\in\!\h_1\,\perp\,\h_2\!
\ni \pi(W(\o a))x_{2,m}\,$ and therefore, for each $\o a$,
$\mu_{x_{2,m}}({S_{1,k}}^{(\o a)})=0$.
Similarly $(\mu_{x_{1,l}}\ot\mu_{x_{2,m}})(\Im_{2,k})=0$
for each $k$ in $I_2$. From Lemma \lpmipr\ it then follows that $x_2=0$.
\qed
The notion of localizability, as introduced by Segal [\seie],
plays an important r\^ole in the following:
a positive measure space $(\Omega, \Sigma, \mu)$ is said to be
{\it localizable} if for every (not necessarily countable)
collection ${\cal G}\subset \Sigma$ of sets of
finite measure there exists a set $B$ in $\Sigma$,
called the {\it supremum}, such that:
(1) for each $E$ in ${\cal G}$, $\mu(E\backslash B)=0$ and
(2) if $\widetilde B$ is another element of $\Sigma$ such
that $\mu(E\backslash \widetilde B)=0$ for every $E$ in ${\cal G}$,
then $\mu(B\backslash \widetilde B)=0$.
\rb
%\def\lmeapr{5.4}
LEMMA \lmeapr. {\it The measure $\mu$ is localizable and
can be written in the form
$$\mu\;=\,\sum_{i\in I}\,\chi_{S_i}\ \mu_{x_0}^{(\o r_i)}
\;\equiv\;\sum_{i\in I}\mu_i$$
where: $\o r_i$'s are points of $\T$, $x_0$ is an
arbitrary non-null vector in $\h$ and ${\{S_i\}}_{i\in I}$ is a
disjoint collection of Borel subsets of $\T$.}
\pf
Let $x_0$ be a fixed non-null vector of $\h$.
For every Borel set $Y$ of $\T$ and every point
$\o r$ in $\T$, define
$$x_{Y,\o r}\,=\,\pi(W(-\o r))\ \,\U^{-1}\chi_Y\,\U\ x_0\ .$$
By Lemma \lawmev,
$\mu_{x_{Y,\o r}}={(\chi_Y\, \mu_{x_0})}^{(\o r)}$.
Let ${\cal J}=\{x_{Y_i,\o r_i}\}_{i\in I}$ be a maximal
family of such vectors, with the properties:
(i) ${\cal J}$ contains $x_0$
(ii) if $x_{Y_i,\o r_i}$ and $x_{Y_j,\o r_j}$ are in ${\cal J}$,
then $\mu_{x_{Y_i,\o r_i}}\! \perp \mu_{x_{Y_j,\o r_j}}$.
>From the orthogonality of these measures it follows that the
corresponding cyclic subspaces $\h_{x_{Y_i,\o r_i}}$ are
mutually orthogonal and the irreducibility of the representation
(Lemma \lirrap) implies that $\h=\oplus_{i\in I} \h_{x_{Y_i,\o r_i}}$.
Then, according to the spectrally multiplicity-free property, we can take
$$\mu\,=\,\sum_{i\in I} {(\chi_{Y_i}\, \mu_{x_0})}^{(\o r_i)}\ .$$
We claim that such a $\mu$, naturally extended to
the $\sigma$-algebra $\bigcap_{x\in\h}\, \sc^{{}_{\mu_{x}}}$ (where
$\sc^{{}_{\mu_{x}}}$ denotes the $\mu_x$-completion of $\bs$) is localizable.
\pn
Firstly we observe that, as a
consequence of the hypotheses (A) and (B), there exists
a Borel set $Y_0$ in $\T$ such that:
(1) $\mu_{x_0}(\T\backslash Y_0)=0$ and $\mu_y(Y_0)=0$ for every
$y\perp \h_{x_0}$
(2) $\{\pi(W(\o a-\o b))\,\h_{x_0}\, |\, \o a\,,\o b\in Y_0\}$
is a separable subset of $\h$.
By the first of these two properties, we can write
$$\mu\,=\, \sum_{i\in I}\,
\chi_{{(Y_i\cap Y_0)}^{(\o r_i)}}\ \mu_{x_0}^{(\o r_i)}\quad .$$
The second property implies that, for each index $i$ in $I$, the set
$$I'\,=\,\bigl \{ j\in I\;\bigl |\; j\ne i\ \hbox {and }\
(Y_0\cap Y_i)^{(\o r_i)} \cap\, (Y_0\cap Y_j)^{(\o r_j)}\,
\ne\emptyset\,\bigr\}$$
is countable. In fact, if
$\o c\in ( (Y_0\cap Y_i)^{(\o r_i)} \,\cap\;
(Y_0\cap Y_j)^{(\o r_j)})$,
then $\o a+\o r_j=\o c=\o b+\o r_i$ for some
$\o a$ in $Y_0\cap Y_j$ and $\o b$ in $Y_0\cap Y_i$;
therefore
$$\bigl\{x_{Y_j,\o r_j}\, \bigr |\, j\in I' \bigr\}\,\subseteq\,
\,\pi(W(-\o r_i))\; \bigl\{\pi(W(\o a-\o b))\,\h_{x_0}\, \bigr |
\, \o a,\o b\in Y_0 \bigr\}$$
and $I'$ must be countable.
In conclusion $\mu$ turns out to be the sum of
finite measures which are concentrated on a
family $\{(Y_0\cap Y_i)^{(\o r_i)}\}_{i\in I}$ of measurable sets such that
$$\mu\,\bigl (\cup_{j\ne i}\,(Y_0\cap Y_j)^{(\o r_j)}\,\cap\,
(Y_0\cap Y_i)^{(\o r_i)} \bigr)
\,=\,\mu\,\bigl (\cup_{j\in I'}\,(Y_0\cap Y_j)^{(\o r_j)}\,\cap\,
(Y_0\cap Y_i)^{(\o r_i)} \bigr)\,=\,0\ .$$
Using this result and the fact each finite measure is localizable,
the localizability of $\mu$ can be easily obtained.
\pn
To conclude the proof of the lemma we note that, since the generators
of $\ccr$ are indexed by $\R^2$ and the representation is cyclic,
the dimension of $\h$ and therefore the cardinality of $I$ cannot exceed
that of continuum.
Since the measure $\mu$ is localizable
and admits the family $\{(Y_0\cap Y_i)^{({\o r}_i)}\,|\, i \in I\}$
of $\mu$-a.e. disjoint measurable sets, with
$00\}$ must be countable (since
$\mu_i\ot\mu_j$ is a finite measure and the
sets $\Im_k$ are disjoint).
Then, if $0\not\in Y$, we write
$${f}^{-1}(Y)\,\cap\, (S_i\times S_j)\,=\,A\,\cup\, B$$
with $A= \cup_{k\in I'}{(f^{(k)} )}^{-1}(Y) \,
\cap\, (S_i\times S_j)$ and $B=\cup_{k\not\in I'}
{(f^{(k)} )}^{-1}(Y) \,\cap\, (S_i\times S_j)$.
The set $A$ is a countable union of
measurable sets. The set $B$ is contained in
${(\cup_{k\in I'}\;\Im_{k} )}^{{\rm c}}$
which is $\mu_i\ot\mu_j$-null. (In fact
$(\mu_i\ot\mu_j)({(\cup_{k\in I'}\;\Im_{k} )}^{{\rm c}} )>0$
would imply, according to Lemma \lpmipr,
the existence of a $k_0$ such that
$(\mu_i\ot\mu_j) ({ (\cup_{k\in I'}\;\Im_{k})}^{{\rm c}}
\cap\, \Im_{k_0} )>0$ and this would imply
$k_0\not\in I'$.) Therefore $B\in \sgp$. If $0\in Y$, the above
argument applies apart from a set contained in
${ (\cup_{k\in I}\,\Im_k)}^{{\rm c}} \cap\, (S_i\times S_j)$
which is $\mu_i\otimes\mu_j$-null.
\pn
Finally for every $k$ in $I$ and $A$ in $\zac$, using the
definition of $T(\o a)$ and the translation property
of the automorphism $\widehat{\tau_{\o a}}$
(see the proof of Lemma \lawmev), one has
$$\eqalign{
\bigl (T(\o a)\; \ta\;\chi_{S_k}\bigr )(\o \a)\,
&=\, \ta(\o \a+\o a)\ \bigl (T(\o a)
\chi_{S_k}\bigr )(\o \a)\cr
&=\, \ta(\o \a+\o a)\ \chi_{S_k}(\o \a+\o a)\
f^{(k)}(\o a, \o \a)\,= \,
f(\o a, \o \a)\ \bigl (\ta\;\chi_{S_k}\bigr )(\o \a+\o a)\cr}$$
and, by continuity,
$\bigl (T(\o a)\psi\bigr )(\o \a)=
f(\o a,\o \a)\, \psi(\o \a+\o a)$ for every
$\psi$ in $\lq (\T ,\mu)$.
\qed
\rb
%\def\lfffat{5.7}
LEMMA \lfffat. {\it There exists a complex function,
$\xi(\o \a)$, on $\T$, measurable w.r.t. the $\sigma$-algebra
$\bigcap_{x\in\h}\, \sc^{{}_{\mu_{x}}}$, such that $|\xi(\o \a)|\in (0,+\infty)\ $
$\mu$-a.e. and, for each $\o a$ in $\T$,
$$f(\o a,\o \a)\;=\;{{\xi(\o a+\o \a)}\over{\xi(\o \a)}}
\qquad\qquad\mu(\o \a)\hbox{-a.e.}\ .$$}
\pf
It is useful to introduce the measure
$$\nu\,=\,\sum_{i,j\in I}\mu_i\ot\mu_j$$
defined on the $\sigma$-algebra $\bigcap_{i,j\in I}\sgp$
and the homeomorphism $\th$ of $\T\times\T$ given by
$\th(\o a,\o \a)=(\o a+\o \a ,\o \a)$.
>From the strong measurability of the representation
and Lemma \lpmipr, it follows that,
if $N\subset \T\times\T$ and $\nu(N)=0$, then also $\nu(\th(N))=0$.
\pn
The group law $T(\o a+\o b)=T(\o a)\circ T(\o b)$ and
Lemma \lfopmt\ imply that, for every $\o a,\o b$ in $\T$,
$$f(\o a+\o b, \o \a)\ =\ f(\o b, \o \a+\o a)\; f(\o a,\o \a)
\qquad\qquad\mu(\o \a)\hbox{-a.e.}$$
and, replacing $f$ with $g=f\circ \th^{-1}$, we obtain
$$g(\o a+\o b+\o \a, \o \a)= g(\o a+\o b+\o \a, \o \a+\o a)\ \,
g(\o a+\o \a,\o \a)\qquad\qquad \mu(\o \a)
\hbox{-a.e.}\ .$$
Thus the function $G$ given by
$$G(\o b,\o a,\o \a)\,=\, g(\o a+\o b+\o \a\,, \o \a) - g(\o a+\o b+\o \a\,,
\a+\o a)\ \,g(\o a+\o \a\,,\o \a)$$
is equal to zero $\nu(\o a,\o \a)$-a.e. for each fixed $\o b$.
According to the properties of $\th$ we have that
$G(\o b, \cdot, \cdot)\circ\th^{-1}$ is still
$\nu(\o a,\o \a)$-a.e. null, i.e.
%\def\efffat{5.7}
$$ g(\o a+\o b\,, \o \a)\,=\, g(\o a+\o b\,, \o a)\ \,g(\o a\,,\o \a)
\qquad\quad \nu(\o a,\o \a)\hbox{-a.e. for each }
\o b \hbox{ in }\T \ . \eqno{(5.1)} $$
The strong measurability of the representation and the
unitarity of $T(\o a)$ ensure that (see [\cath]) there exist at least a
Borel set $X_0$ in $\T$ such that the function
$$\xi\bigl (\o b\bigr)\, =
\,\int_{X_0} g(\o b,\o \a)\ \d\mu_{x_0}(\o \a)$$
is well-defined and such that
$|\xi\bigl(\o b\bigr)|\in (0,+\infty)$
$\mu$-almost everywhere; furthermore $\xi$ is measurable
w.r.t. the $\sigma$-algebra $\bigcap_{x\in \h}\sc^{{}_{\mu_x}}$.
Then, using relation (5.1), we can conclude that,
for each $\o d$ in $\T$,
$$g(\o b+\o d,\o b)\ \,\xi\bigl (\o b\bigr)\,=
\int_{X_0}\!\! g(\o b +\o d,\o \a)\ \d\mu_{x_0}(\o \a)\,=
\,\xi\bigl(\o b+\o d\bigr)$$
$\mu(\o b)$-almost everywhere.
So, for every $\o d$ in $\T$, we have
$$\qquad\qquad f(\o d,\o b)\, =\, g(\o b+\o d,\o b)\,=\,
{{\xi\bigl(\o b+\o d\bigr)}\over{\xi\bigl(\o b\bigr)}}
\qquad\qquad\qquad\mu(\o b)\hbox{-a.e.}$$
and the lemma is proved.
\qed
To complete the proof of the theorem
define now a new measure $\widetilde\mu$ on $\T$ setting
$$\widetilde\mu\, =\, |\xi|^{-2}\, \mu\ .$$
It is not difficult to verify that the map
${\mit\Xi}$, given by
$({\mit\Xi}\psi)(\a,\b) = \xi(\a,\b)\ \psi(\a,\b)$,
is a norm-preserving isomorphism
from $\lq(\T,\mu)$ onto $\lq(\T,\widetilde\mu)$ and,
for each $\widetilde \psi$ in
$\lq(\T,\widetilde\mu)$ and each $(a,b)$ in $\T$,
$$\bigl ({\mit\Xi}\; T(a,b)\;{\mit\Xi}^{-1}\ \widetilde\psi\,
\bigr)(\a,\b)\,=
\,\widetilde\psi((\a+a)\mod 1,(\b+b)\mod 2\pi)\ .$$
This equation implies that
$\widetilde\mu$ is translation-invariant
(since the operators
${\mit\Xi}\;T(a,b)\;{\mit\Xi}^{-1}$
are norm-preserving) and the isomorphism
${\mit\Xi}\circ\U$ (from $\h$
onto $\lq(\T,\widetilde \mu)$) gives equations
(4.1) for the action of the Weyl operators.
%
\rc\rc
%
\vfill\eject
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\end